The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(0, 1, x) → f(x, x, x)
f(x, y, z) → 2
0 → 2
1 → 2

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(z0, z0, z0)
f(z0, z1, z2) → 2
0 → 2
1 → 2

Tuples:

F(0, 1, z0) → c(F(z0, z0, z0))

S tuples:

F(0, 1, z0) → c(F(z0, z0, z0))

K tuples:none
Defined Rule Symbols:

f, 0, 1

Defined Pair Symbols:

F

Compound Symbols:

c

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

F(0, 1, z0) → c(F(z0, z0, z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(z0, z0, z0)
f(z0, z1, z2) → 2
0 → 2
1 → 2

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, 0, 1

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))